INTERPOLATING BLASCHKE PRODUCTS AND ANGULAR DERIVATIVES

We show that to each inner function, there corresponds at least one interpolating Blaschke product whose angular derivatives have precisely the same behavior as the given inner function. We characterize the Blaschke products invertible in the closed algebra H-infinity[(b) over bar : b has finite angular derivative everywhere. We study the most well-known example of a Blaschke product with infinite angular derivative everywhere and show that it is an interpolating Blaschke product. We conclude the paper with a method for constructing thin Blaschke products with infinite angular derivative everywhere.

Interpolating Blaschke Products and Angular Derivatives

We show that to each inner function, there corresponds at least one interpolating Blaschke product whose angular derivatives have precisely the same behavior as the given inner function. We characterize the Blaschke products invertible in the closed algebra generated by the algebra of bounded analytic functions and the conjugates of Blaschke products with angular derivative finite everywhere. We study the most well-known example of a Blaschke product with infinite angular derivative everywhere and show that it is an interpolating Blaschke product. We conclude the paper with a method for constructing thin Blaschke products with infinite angular derivative everywhere.

Toluene dioxygenase-catalysed oxidation route to angular cis-monohydrodiols and other bioproducts from bacterial metabolism of 1,2-dihydrobenzocyclobutene and derivatives

A mutant strain (UV4) of the soil bacterium Pseudomonas putida, containing toluene dioxygenase, has been used in the metabolic oxidation of 1,2-dihydrobenzocyclobutene 12 dagger and the related substrates 1,2-dihydrobenzocyclobuten-1-ol 13 and biphenylene 33. Stable angular cis-monohydrodiol metabolites (1R,2S)-bicyclo[4.2.0]octa-3,5-diene-1,2 7, (1S,2S,8S)-bicyclo[4.2.0]octa-3,5-diene-1,2,8-triol 8 and biphenylene-cis-1,8b-diol 9, isolated from each of these substrates, have been structurally and stereochemically assigned. The structure, enantiopurity and absolute configuration of the other cis-diol metabolites, (2R,3S)-bicyclo[4.2.0]octa-1(6),4-diene-2,3-diol 14 and cis-1,2-dihydroxy-1,2-dihydrobenzocyclobutene 16, and the benzylic oxidation bioproducts, 1,2-dihydrobenzocyclobuten-1-ol 13, 1,2-dihydrobenzocyclobuten-1-one 15 and 2-hydroxy-1,2-dihydrobenzocyclobuten-1-one 17, obtained from 1,2-dihydrobenzocyclobutene and 1,2-dihydrobenzocyclobuten-1-ol, have been determined with the aid of chiral stationary-phase HPLC, NMR and CD spectroscopy, and stereochemical correlation. X-Ray crystallographic methods have been used in the determination of absolute configuration of the di-camphanates 27 (from diol 7) and 32 (from diol 9), and the di-MTPA ester 29 (from diol 14) of the corresponding cis-diol metabolites. The metabolic sequence involved in the formation of bioproducts derived from 1,2-dihydrobenzocyclobutene 12 has been investigated.

Integrating multibeam backscatter angular response, mosaic and bathymetry data for benthic habitat mapping

Multibeam echosounders (MBES) are increasingly becoming the tool of choice for marine habitat mapping applications. In turn, the rapid expansion of habitat mapping studies has resulted in a need for automated classification techniques to efficiently map benthic habitats, assess confidence in model outputs, and evaluate the importance of variables driving the patterns observed. The benthic habitat characterisation process often involves the analysis of MBES bathymetry, backscatter mosaic or angular response with observation data providing ground truth. However, studies that make use of the full range of MBES outputs within a single classification process are limited. We present an approach that integrates backscatter angular response with MBES bathymetry, backscatter mosaic and their derivatives in a classification process using a Random Forests (RF) machine-learning algorithm to predict the distribution of benthic biological habitats. This approach includes a method of deriving statistical features from backscatter angular response curves created from MBES data collated within homogeneous regions of a backscatter mosaic. Using the RF algorithm we assess the relative importance of each variable in order to optimise the classification process and simplify models applied. The results showed that the inclusion of the angular response features in the classification process improved the accuracy of the final habitat maps from 88.5% to 93.6%. The RF algorithm identified bathymetry and the angular response mean as the two most important predictors. However, the highest classification rates were only obtained after incorporating additional features derived from bathymetry and the backscatter mosaic. The angular response features were found to be more important to the classification process compared to the backscatter mosaic features. This analysis indicates that integrating angular response information with bathymetry and the backscatter mosaic, along with their derivatives, constitutes an important improvement for studying the distribution of benthic habitats, which is necessary for effective marine spatial planning and resource management.

Numerical simulation for the 3D seepage flow with fractional derivatives in porous media

In this paper, the numerical simulation of the 3D seepage flow with fractional derivatives in porous media is considered under two special cases: non-continued seepage flow in uniform media (NCSFUM) and continued seepage flow in non-uniform media (CSF-NUM). A fractional alternating direction implicit scheme (FADIS) for the NCSF-UM and a modified Douglas scheme (MDS) for the CSF-NUM are proposed. The stability, consistency and convergence of both FADIS and MDS in a bounded domain are discussed. A method for improving the speed of convergence by Richardson extrapolation for the MDS is also presented. Finally, numerical results are presented to support our theoretical analysis.

Modified alternating direction methods for solving a two-dimensional non-continuous seepage flow with fractional derivatives

In this paper, a two-dimensional non-continuous seepage flow with fractional derivatives (2D-NCSF-FD) in uniform media is considered, which has modified the well known Darcy law. Using the relationship between Riemann-Liouville and Grunwald-Letnikov fractional derivatives, two modified alternating direction methods: a modified alternating direction implicit Euler method and a modified Peaceman-Rachford method, are proposed for solving the 2D-NCSF-FD in uniform media. The stability and consistency, thus convergence of the two methods in a bounded domain are discussed. Finally, numerical results are given.

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order αset membership, variant(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order βset membership, variant(0,1) and of order αset membership, variant(1,2], respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.

Do derivatives have a role in the risk-shifting behaviour of fund managers?

In this paper we examine the extent to which derivatives are used to affect the risk-shifting behaviour of Australian equity fund managers. We find, after periods of good and poor performance, the risk-shifting behaviour of fund managers is different between derivative users and non-users. Our results support the gaming and active competition hypotheses but there is little support for the cash flow hypothesis. The study also allows for a complex reporting environment by analysing data across three alternate time periods: the calendar year, financial year and quarterly frames. Given that our results are not consistent across time periods for users and non-users of derivatives, some caution in interpretation is required.

Estimating the loading condition of a diesel engine using instantaneous angular speed analysis

Continuing monitoring of diesel engine performance is critical for early detection of fault developments in the engine before they materialize and become a functional failure. Instantaneous crank angular speed (IAS) analysis is one of a few non intrusive condition monitoring techniques that can be utilized for such tasks. In this experimental study, IAS analysis was employed to estimate the loading condition of a 4-stroke 4-cylinder diesel engine in a laboratory condition. It was shown that IAS analysis can provide useful information about engine speed variation caused by the changing piston momentum and crankshaft acceleration during the engine combustion process. It was also found that the major order component of the IAS spectrum directly associated with the engine firing frequency (at twice the mean shaft revolution speed) can be utilized to estimate the engine loading condition regardless of whether the engine is operating at normal running conditions or in a simulated faulty injector case. The amplitude of this order component follows a clear exponential curve as the loading condition changes. A mathematical relationship was established for the estimation of the engine power output based on the amplitude of the major order component of the measured IAS spectrum.